Q.E.D.

42 43 the numbers oF nature the geometry of growth A spiral of squares growing around a unit square as on the top left consists of squares the lengths of whose sides are the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, ..., named after Leonardo Fibonacci 1170 1250. Every number in the sequence is the sum of the two preceding it, so that 2 1 1, 3 1 2, 5 2 3, and so on. Fibonacci numbers are connected in many wonderful ways. For example, the tiling of the rectangle on the top left demonstrates that 12 12 22 32 52 82 132 13.13 8 13.21. In general, the sum of the squares of the first n Fibonacci numbers equals the product of the nth and n1st such numbers. Similarly, the tiling of the square on the right shows that 1.1 2.1 3.2 5.3 8.5 13.8 21.13 212. This equality also generalises easily. The Fibonacci numbers often show up in the same phenomena as the golden ratio f see previous page and it can be proved that the nth Fibonacci number is the closest natural number to f n5. This implies that the rectangles we come across when building our spiral of squares become indistinguishable from golden rectangles. Fibonacci numbers are hidden in many growth processes. For example, the numbers of clockwise and counterclockwise spirals apparent in sunflower heads opposite, middle are usually consecutive Fibonacci numbers. Pascals triangle lower, opposite also grows, here row by row, with neighboring entries in one row adding up to the number below them. Since the sums of the first two diagonals of this triangle are both 1, and the sums of any two consecutive diagonals add up to the sum of the next, our golden sequence will appear yet again.